a study of the vibration signals behaviour lim jun lun...

A STUDY OF THE VIBRATION SIGNALS BEHAVIOUR

LIM JUN LUN

Thesis submitted in fulfilment of the requirements

for the award of the degree of

Bachelor of Mechanical Engineering

Faculty of Mechanical Engineering

UNIVERSITI MALAYSIA PAHANG

JUNE 2013

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ABSTRACT

Identification of defects in structures and its components is a crucial aspect in

decision making about their repair and replacement. This project presents a study on the

application of free and force vibration signals on difference surface beam to detect

resonance frequency or natural frequency and the statistical characteristics. The well-

established statistical parameters such as kurtosis, mean value, standard deviation, RMS

and variance are utilized in the study. Free and force vibration experiments are to

determine the natural frequency of the beam from the vibration signal and the vibration

signal is use to be analysis on statistical analysis. Under ideal conditions, the modal

testing method can be used to identify the natural frequency and mode shape, because it

more accurate compares to free and force vibration experiment. The effect of surface

beam on the performance of the statistical method is also studied. Results from the

study show that the statistical parameters are affected by the surface beam due to the

change of the beam structure of which have difference natural frequencies

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ABSTRAK

Pengenalan kecacatan dalam struktur dan komponen merupakan aspek yang

penting dalam membuat keputusan mengenai pembaikian dan penggantian. Projek ini

membentangkan kajian atas getaran bebas and paksa terhadap pelbagai permukaan

rasuk untuk mengesan frekuensi, resonans dan ciri-ciri statistic. Parameter statistik

seperti kurtosis, nilai min, sisihan piawai, RMS dan varians digunakan dalam kajian ini.

Eksperimen getaran bebas and paksa adalah untuk menentukan rasuk frekuensi

resonans daripada isyarat getaran dan isyarat getaran ini akan gunakan untuk menjadi

analisis statistik. Dalam kaedah ujian mod boleh digunakan untuk mengenal pasti

frekuensi resonans dan bentuk mod, disebabkan ia lebih tepat dibandingkan dengan

getaran bebas dan paksa. Kesan daripada permukaan rasuk bagi statistik juga dikajikan.

Hasil daripada kajian menunjukkan bahawa parameter statistik dipengaruhi oleh

permukaan rasuk, hal ini disebabkan oleh perubahan rasuk struktur yang mempunyai

frekuensi resonans yang berbezaan.

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TABLE OF CONTENTS

Page

SUPERVISOR’S DECLARATION ii

STUDENT’S DECLARATION iii

ACKNOWLEDGEMENTS iv

ABSTRACT v

ABSTRAK vi

TALBE OF CONTENTS vii

LIST OF TABLES x

LIST OF FIGURES xi

LIST OF SYNBOLS xii

LIST OF ABBREVIATIONS xiii

CHAPTER 1 INTRODUCTION

1.0 Introduction 1

1.1 Background of Study 1

1.2 Problem Statement 2

1.3 Objectives 3

1.4 Scopes 3

CHAPTER 2 LITERATURE REVIEW

2.0 Introduction 4

2.1 Signal 4

2.1.1 Random 5

2.1.2 Stationary 5

2.1.3 Non-Stationary Signal 5

2.1.4 Deterministic Signal 5

2.2 Free Vibration 6

2.3 Forced Vibration 8

2.4 Free and Forced Vibration on Cracked Beam 10

2.5 Frequency and Time Domain 12

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2.6 Statistical Analysis 14

2.6.1 Statistics Parameter 14

2.7 Modal Testing 16

2.8 Conclusion 17

CHAPTER 3 METHODOLOGY

3.0 Introduction 18

3.1 Fabrication 19

3.2 Experimental 20

3.2.1 Forced Vibration 20

3.2.2 Free Vibration 21

3.2.3 Modal Testing 21

3.2.4 Software Specification 22

3.2.5 Hardware Specification 23

3.3 Analysis 23

CHAPTER 4 RESULT AND DISCUSSION

4.0 Introduction 24


4.2 Force Vibration 26

4.3 Free Vibration 28

4.4 Force Vibration Statistical Parameter Analysis 31

4.5 Free Vibration Statistical Parameter Analysis 34

CHAPTER 5 CONCLUSION AND RECOMMENDATION

5.0 Conclusion 37

5.1 Recommendation 38

REFERENCES 39

APPENDICES

A Gantt chart for FYP 1 and FYP 2 41

ix

LIST OF TABLES

Table No. Title Page

3.1 Software Specification for Vibration Analysis 22

3.2 Hardware Specification for Vibration Analysis 23

4.1 Depth of the Crack Beam 24

4.2 Resonance Frequency on 1st Mode 25

4.3 Minimum and Maximum of Vibration Amplitude 27

4.4 Variance between Force Vibration and Modal Testing 28

4.5 Frequency of Beam 1 29



4.8 Variance between Free Vibration and Modal Testing 30

4.9 Comparison Variance between Force and Free Vibration 30

4.10 Statistical Value for Beam 1 31



4.13 Summary overall Force Vibration Statistical Value 32




4.17 Summary overall Free Vibration Statistical Value 35

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LIST OF FIGURES

Figure No. Title Page

2.1 Steady-state mode 6

2.2 Steady-state and transient 10

2.3 Distribution with too much peak 15

2.4 Too flat distribution 16

3.1 Flow Chart 18

3.2 Smooth Beam 19

3.3 Crack Beam 19

3.4 Experiment Setup 20


4.1 1st Mode Shape 25

4.2 Resonance Frequency versus Depth of Crack 26

4.3 Vibration Signal Beam 1 at 6Hz 26

4.4 Amplitude verses Frequency 27

4.5 Free Vibration Signal Beam 1 28

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LIST OF SYMBOLS

Mean

Root mean square

Kurtosis

Phase shift

natural frequency

damping ratio

xii

LIST OF ABBREVIATIONS

DUT Device under test

FFT The fast fourier transform

DFT Discrete fourier transforms

IDFT Inverse discrete fourier transforms

RMS Root mean square

CHAPTER 1

INTRODUCTION

1.0 INTRODUCTION

This chapter explains about the background of study, problem statement,

objectives and the scopes of this study. The main purpose for this study can be

identified by referring at the problem statement of this study. Furthermore, the details of

this study and outcome can be achieved on the objectives and its scopes.

1.1 BACKGROUND OF STUDY

Vibration can be mention as the motion of a subject or a body or system of

connected bodies and displaced from a position of equilibrium. Some of the vibrations

are phenomena that undesirable in machines or structures in our reality life because

vibrations can produce stresses, causes the system to lose energy, tools will easily wear

and tear. Vibration occurs when a system is displaced from a position of stable

equilibrium. Which mean that a system is under the restoring forces such as the elastic

forces, gravitational forces like a simple pendulum keeps moving back and forth across

its position of equilibrium and at the end system is able to return to the it’s equilibrium

position. A system is a combination of varies element intended to work with each other

to fulfill an object. An automobile like a car is a good example of system whose

elements are the wheels, suspension, seat, and car body.

Vibration mostly can divide into a two major part which is free and forced

vibration. The different between the two parts can be easy recognizing. Free vibration

can be simply represent the natural frequency of a system example the vibration of a

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pendulum is free vibration which needs no external force to vibrate. While for a forced

vibration can be explained when a periodically varying force is applied on a system, the

system will vibrate in accordance to the frequency of applied force. The frequency of

vibration of the system is the same as the frequency of the applied force.

Vibration signal can be obtain by using vibration transducers which is

accelerometers, velocity transducers and displacement transducers. Each vibration

transducer has its own specific characteristics and use in different applications depend

on the process of vibration analysis. Accelerometers are the famous and most widely

used vibration transducers for measuring vibrations on stationary machine elements. An

accelerometer is a full-contact transducer mounted directly on a system or device under

test (DUT). When a system is vibrate or change in motion will produce a force that

causes the mass inside the sensor to "squeeze" the piezoelectric which produces an

electrical charge that is proportional to the force exerted upon it. Base from the equation,

the charge produce by the piezoelectric is proportional to the force, while the

mass is a constant for anytime, and the charge is proportional to the acceleration. The

benefits of an accelerometer include linearity over a wide frequency range and a large

dynamic range can be using most accelerometers in hazardous environments because of

their rugged and reliable construction.

1.2 PROBLEM STATEMENT

Mostly of the structures or an object are weakened by cracks. When the crack

size increases in course of time, the structure becomes weaker than its previous

condition. Finally, the structure may breakdown due to a minute crack. Therefore, crack

detection and signal vibrate for classification is a very important issue. In this study,

free and forced vibration analysis of a cracked beam was performed in order to identify

the crack situation and the signal of vibration compare with uncrack beam. Structures is

required that must be good in condition during its service life that to confirm that the

environment is under guarded and safety. Cracks are among the most encountered

damage types in the structures. Cracks in a structure may be hazardous due to static or

dynamic loadings, so that study on crack vibration signal plays an important role for

structural health monitoring applications.

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1.4 OBJECTIVES

The objectives of this project are:

i. To study the natural frequency of the free and forced vibration signals with

difference depth of crack.

ii. To study the statistical analysis of the free and forced vibration signals with

difference depth of crack.

1.4 SCOPE

The scopes of this project are:

i. Perform experimental testing for the data measurement purposes

ii. Perform statistical analysis and frequency-time domain analysis

iii. Signal classification base on the value of the statistical parameter

CHAPTER 2

LITERATURE REVIEW

2.0 INTRODUTION

This chapter is a review of the literature that discusses to identify studies

relevant to the topic. The purpose of this chapter is to know more about the detail of the

cracked beam vibration signal under the free and forced axial loading analysis by using

DASY LAB software. There are three categories discuss in this chapter: type of

vibration, frequency and time domain and the statistical analysis. The first topic

explains about the signal. The second topic explains about variable forces and force act

on the cracked beam. The third topic explains about the frequency on the reaction beam

when it vibrates. Lastly fourth topic explains about the statistical analysis that used to

classify the signal based on the statistical parameter.

2.1 SIGNAL

Signal is detectable transmitted energy that can be used to carry information.

Beside signal also is time-dependent variation of a characteristic of a physical

phenomenon, used to convey information. Signal can as applied to electronics, any

transmitted electrical impulse. Operationally, signal is a type of message, the text of

which consists of one or more letters, words, characters, signal flags, visual displays, or

special sounds, with prearranged meaning and which is conveyed or transmitted by

visual, acoustical, or electrical means. The measured signal is usually converted into an

electric signal, which it passed through a series of electrical or electronic circuits to

achieve the required processing. (Leonhard, 1999)

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2.1.1 Random

Random signals are random variables which evolve, often with time, but also

with distance or sometimes another parameter. White noise is a random signal or

process with a flat power spectral density. In other words, the signal contains equal

power within a fixed bandwidth at any centre frequency. Random signals are

unpredictable in their frequency content and their amplitude level, but they still have

relatively uniform statistical characteristics over time. Examples of random signals are

rain falling on a roof, jet engine noise, turbulence in pump flow patterns and cavitation.

(Shanmugam & Breiphol, 1988)

2.1.2 Stationary

The first natural division of all signals is into either stationary or non-stationary

categories. Stationary signals are constant in their statistical parameters over time. If

look at a stationary signal for a few moments and then wait an hour and look at it again,

it would look essentially the same. Rotating machinery generally produces stationary

vibration signals. (Groepner, 1991)

2.1.3 Non-Stationary Signals

Non-stationary signals are divided into continuous and transient types. Transient

signals are defined as signals which start and end at zero level and last a finite amount

of time. They may be very short, or quite long. Examples of transient signals are a

hammer blow, an airplane flyover noise, or a vibration signature of a machine run up or

run down. (Crocker & Malcom, 1998)

2.1.4 Deterministic Signal

Deterministic signals are a special class of stationary signals, and they have a

relatively constant frequency and level content over a long time period. Deterministic

signals are generated by rotating machines, musical instruments, and electronic function

generators. They are further divisible into periodic and quasi-periodic signals. Periodic

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signals have waveforms whose pattern repeats at equal increments of time, whereas

quasi-periodic signals have waveforms whose repetition rate varies over time, but still

appears to the eye to be periodic. Sometimes, rotating machines will produce quasi-

periodic signals, especially belt-driven equipment. (Temme, 1998)

2.2 FREE VIBRATION

The simplest natural vibration system, having mass and a linear stiffness element,

in a time domain gives rise to pure harmonic motion. Free vibration occurs when a

mechanical system is set off with an initial input and then allowed to vibrate freely. Real

free vibration always gradually decreases in amplitude due to system energy losses.

Free vibration also is the natural response of a structure to some impact or displacement.

The response is completely determined by the properties of the structure, and its

vibration can be understood by examining the structure's mechanical properties.

(Feldmen, 1985)

For the steady state (solid line), the signal of vibration is in steady mode, smooth

and average show in figure 2.1. Steady state vibration exists in a system if the velocity

is a continuous periodic quantity. (Lin & Hsiao, 2001)

Figure 2.1: Steady State

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The questions for free vibration without damping:

(2.1)

is represent as the force generated by the system, is represent the stiffness of the

spring and is represent the amplitude of the vibration.

Amplitude also can be derived as below:

( ) ( ) (2.2)

is represent by maximum amplitude of the vibration system, is represent by the un-

damped natural frequency and is represent the vibration time.

Natural frequency also can be derived as below:

√

(2.3)

is represent by the stiffness and is represent the mass. (Chen & Yang, 1985)

The equations for free vibration with damping:

(2.4)

is represent the damping coefficient and is represent the velocity.

The equations for un-damped natural frequency:

√ (2.5)

is represent the natural frequency and ζ is represent the damping ratio

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ζ also can derive as below:

√ (2.6)

is represent the damping coefficient, is represent by the stiffness and is represent

the mass.

The damped natural frequency is less than the un-damped natural frequency, but

for many practical cases the damping ratio is relatively small and hence the difference is

negligible. Therefore the damped and un-damped description is often dropped when

stating the natural frequency. (Hagood & Flotow, 1991)

2.3 FORCED VIBRATION

Forced vibration is the response of a structure to a repetitive forcing function

that causes the structure to vibrate at the frequency of the excitation. In forced vibration

there is a relationship between the amplitude of the forcing function and the

corresponding vibration level. The relationship is dictated by the properties of the

structure. Forced vibration is when an alternating force or motion is applied to a

mechanical system. (Oniszczuk, 2003)

The equations for forced vibration with damping:

( ) (2.7)

Sum the forces on the mass:

( ) (2.8)

The steady state:

( ) ( ) (2.9)

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The result states that the mass will oscillate at the same frequency, of the applied

force, but with a phase shift, .

The amplitude of the vibration is defined by the following formula.

( ) ( ) (2.10)

Where r is represent as the ratio of the harmonic force frequency over the un-damped

natural frequency of the mass-spring-damper model.

(2.11)

The phase shift , is defined by following formula.

(

) (2.12)

Transient vibration is defined as a temporarily sustained vibration of a

mechanical system. It may consist of forced or free vibrations, or both. Transient

loading, also known as impact, or mechanical shock, is a non-periodic excitation, which

is characterized by a sudden and severe application. For the figure 2.2 of forced

vibration, the signal of vibration is not steady at the start but will become stable after a

period. (Timoshenko & Young, 1974)

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Figure 2.2: Steady State and Transient

2.4 FREE AND FORCED VIBRATION ON CRACKED BEAM

Kim and Zhao (2004) proposed a novel crack detection method using harmonic

response. In their method, displacement and slope modes of a cracked cantilever beam

are considered first, and then the approximate formula for displacement and slope

response under single-point harmonic excitation is derived. They conclude that the slope

response has a sharp change with the crack location and the depth of crack, and

therefore, it can be used as a crack detection criterion. Ruotolo et al (1996) investigated

forced response of a cantilever beam with a crack that fully opens or closes, to

determine depth and location of the crack. In their study, left end of the beam is

cantilevered and right end is free. The harmonic sine force was applied on the free end

of the beam. Vibration amplitude of the free end of the beam was taken into

consideration. It was shown that vibration amplitude changes, when depth and location

of the crack change.

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Pugno and Surace (2000) analyzed the response to harmonic sinusoidal force of

a cantilever beam with several breathing cracks of different size and location, using a

harmonic balance method. Their conclusion is that the presence of breathing cracks in a

beam results in a nonlinear dynamic behaviour which gives rise to super harmonics in

the spectrum of the response signals, the amplitude of which depends on the number,

location and depth of any cracks present. Moreover, they stated that the harmonic

balance method reduces the computation times by approximately 100 times compared to

direct numerical integration.

The free bending vibrations of an Euler-Bernoulli beam of a constant rectangular

cross-section are given by the following differential equation as given in below:

(2.13)

Where m is the mass of the beam per unit length (kg/m) is the natural

frequency of the th mode (rad/s), E is the modulus of elasticity (N/m2) and, I is the

area moment of Inertia (m4).

By defining

is rearranged as a fourth-order differential equation as follows:

(2.14)

The general solution is:

(2.15)

Where A, B, C, D are constants and is a frequency parameter. Since the bending

vibration is studied, edge crack is modeled as a rotational spring with a lumped stiffness.

The crack is assumed open. Based on this modeling, the beam is divided into two

segments: the first and second segments are left and right-hand side of the crack,

respectively. When this equation is solved by applying beam boundary conditions and

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compatibility relations, the natural frequency of the th mode for un-cracked (2.16) and

cracked beams (2.17) is finally obtained.

√

(2.16)

√

(2.17)

Where is the th mode frequency of the un-cracked beam and is a known constant

depending on the mode number and beam end conditions. is the h mode frequency

of the cracked beam, is the ratio between the natural frequencies of the cracked and

un-cracked beam. L is length of the beam. (Dado, 1997)

2.5 FREQUENCY AND TIME DOMAIN

Frequency analysis also provides valuable information about structural vibration.

Any time history signal can be transformed into the frequency domain. The most

common mathematical technique for transforming time signals into the frequency

domain is called the Fourier Transform, after the French Mathematician Jean Baptiste

Fourier. The math is complex, but today's signal analyzers race through it automatically,

in real-time. Fourier Transform theory says that any periodic signal can be represented

by a series of pure sine tones. In structural analysis, usually time waveforms are

measured and their Fourier Transforms computed. The Fast Fourier Transform (FFT) is

a computationally optimized version of the Fourier Transform. (Ratcliffe, 1997)

The equation for Discrete Fourier transforms (DFT):

( ) ∑ ( )

, (2.18)

(2.19)

The equation for Inverse Discrete Fourier transforms (IDFT),

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( )

∑ ( )

, (2.20)

Time domain analysis starts by analysing the signal as a function of time. An

oscilloscope, data acquisition device, or signal analyser can be used to acquire the signal.

The plot of vibration versus time provides information that helps characterize the

behaviour of the structure. It behaviour can be characterized by measuring the

maximum vibration or the peak level, or finding the period time between zero crossings.

(Senra & Correa, 2002)

Loutridis (2005) applied the instantaneous frequency and empirical mode

decomposition methods for crack detection in a cantilever beam with single breathing

crack, example, opens and closes during vibration. They investigated the dynamic

behaviour of the beam under harmonic excitation, both, theoretically and experimentally.

A single degree-of-freedom mathematical model with varying stiffness is employed to

simulate the dynamic response of the beam. They modelled the time varying stiffness

using a simple periodic function. Both simulated and experimental response data are

analysed by applying empirical mode decomposition and the Hilbert transform. By this

way, the instantaneous frequency of each oscillatory mode is obtained. They concluded

that the variation of the instantaneous frequency increases with increasing crack depth

and the harmonic distortion increases with crack depth following definite trends and can

also be used as an effective indicator for crack size. They also stated that the proposed

time-frequency approach is superior compared to Fourier analysis.

Differences in natural frequencies for different crack depth and location have

been used for crack detection for a long time. It is reported that the natural frequency

shift is not very sensitive to cracks. For example, a modal frequency shift of less than 5%

was obtained for a crack depth of 50% of the structure thickness. Thus, researchers have

been studied alternative methods. The harmonic response analysis is one of the

alternative methods. (Han, 2005).

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2.6 STATISTICAL ANALYSIS

Statistics is a mathematical science and study of how to collect, organize,

analysis, interpreted, or in the way to present a set of data. Basically statistics can be

consider in mathematical body of science pertaining to the collection, analysis,

interpretation or explanation, and presentation of data (Moses & Lincoln, 1986), while

on the other hand, statistics may also consider it a branch of mathematics (Hays &

William, 1973) concerned with collecting and interpreting data. Statistics usually

considered being a distinct mathematical science rather than a branch of mathematics,

because of its empirical roots and its focus on applications. (Moore & David, 1992)

Statistics is closely related to the probability theory, with which it is often

grouped and the difference is roughly that in probability theory, one starts from the

given parameters of a total population to deduce probabilities pertaining to samples, but

statistical inference moves in the opposite direction, inductive inference from samples to

the parameters of a larger or total population.

2.6.1 STATISTICS PARAMETER

In statistics, mean can define as arithmetic mean of a sample also called the un-

weighted average. Equation for mean:

∑

(2.21)

is represent mean, is represent the total number of the data in a set of values, and

is represent the data. The mean is the minimum point, by dividing a series of numbers in

to an average point (Feller & William, 1950)

The root mean square also defines as the quadratic mean. The root mean square

is a statistical measure of the magnitude of a set of numbers. Equation for RMS:

√∑

(2.22)

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is represent root mean square, is represent the total number of the data in a set of

values, and is represent the data. The root mean square is square root of the average

of the squares of a variable quantity. (Cartwright & Kenneth, 2007).

Kurtosis characterizes the relative peakedness or flatness of a distribution

compared to the normal distribution. Equation of kurtosis:

∑ ( )

( ) (2.23)

is represent mean, is represent the total number of the data in a set of values, and

is represent the data, is standard deviation (Kevin & MacGillivray, 1988). Positive

kurtosis indicates a relatively peaked distribution (figure 2.3). Negative kurtosis

indicates a relatively flat distribution (figure 2.4)

Figure 2.3: Distribution with Too Much Peak (Kurtosis > 0)

a study of the vibration signals behaviour lim jun lun...

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